Infinite products - reference needed!
I will answer your question
"Most importantly I'd like to know why $$ \prod (1+|a_n|) \to a < \infty \quad \Longrightarrow \quad \prod (1+ a_n) \to b \neq 0. "$$
We will first prove that if $\sum \lvert a_n \rvert < \infty$, then the product $\prod_{n=1}^{\infty} (1+a_n)$ converges. Note that the condition you have $\prod (1+|a_n|) \to a < \infty$ is equivalent to the condition that $\sum \lvert a_n \rvert < \infty$, which can be seen from the inequality below. $$\sum \lvert a_n \rvert \leq \prod (1+|a_n|) \leq \exp \left(\sum \lvert a_n \rvert \right)$$
Further, we will also show that the product converges to $0$ if and only if one of its factors is $0$.
If $\sum \lvert a_n \rvert$ converges, then there exists some $M \in \mathbb{N}$ such that for all $n > M$, we have that $\lvert a_n \rvert < \frac12$. Hence, we can write $$\prod (1+a_n) = \prod_{n \leq M} (1+a_n) \prod_{n > M} (1+a_n)$$ Throwing away the finitely many terms till $M$, we are interested in the infinite product $\prod_{n > M} (1+a_n)$. We can define $b_n = a_{n+M}$ and hence we are interested in the infinite product $\prod_{n=1}^{\infty} (1+b_n)$, where $\lvert b_n \rvert < \dfrac12$. The complex logarithm satisfies $1+z = \exp(\log(1+z))$ whenever $\lvert z \rvert < 1$ and hence $$ \prod_{n=1}^{N} (1+b_n) = \prod_{n=1}^{N} e^{\log(1+b_n)} = \exp \left(\sum_{n=1}^N \log(1+b_n)\right)$$ Let $f(N) = \displaystyle \sum_{n=1}^N \log(1+b_n)$. By the Taylor series expansion, we can see that $$\lvert \log(1+z) \rvert \leq 2 \lvert z \rvert$$ whenever $\lvert z \rvert < \frac12$. Hence, $\lvert \log(1+b_n) \rvert \leq 2 \lvert b_n \rvert$. Now since $\sum \lvert a_n \rvert$ converges, so does $\sum \lvert b_n \rvert$ and hence so does $\sum \lvert \log(1+b_n) \rvert$. Hence, $\lim_{N \rightarrow \infty} f(N)$ exists. Call it $F$. Now since the exponential function is continuous, we have that $$\lim_{N \to \infty} \exp(f(N)) = \exp(F)$$ This also shows that why the limit of the infinite product $\prod_{n=1}^{\infty}(1+a_n)$ cannot be $0$, unless one of its factors is $0$. From the above, we see that $\prod_{n=1}^{\infty}(1+b_n)$ cannot be $0$, since $\lvert F \rvert < \infty$. Hence, if the infinite product $\prod_{n=1}^{\infty}(1+a_n)$ is zero, then we have that $\prod_{n=1}^{M}(1+a_n) = 0$. But this is a finite product and it can be $0$ if and only if one of the factors is zero.
Most often this is all that is needed when you are interested in the convergence of the product expressions for the $L$ functions.
I am making this CW, so that other people can add further references.
Books
Konrad Knopp: Infinite Sequences and Series; see p.92. I believe Knopp's books can be considered a classical references for this.
Konrad Knopp: Theorie und Anwendung der unendlichen Reihen; or the English translation Theory and application of infinite series. There is a whole chapter devoted to infinite products, see p.218. The book is freely available here. This is given as reference in Wikipedia article. (You can read this on Talk page of that article: This article is probably very non-ideal, but when I needed this material a while ago it was hard to find, and so I figured wikipedia would be a good place to hold it. So it seems you are not the there are other people who had problems with finding references about infinite products.)
Earl David Rainville: Infinite Series. This book was mentioned in connection with infinite products in this answer.
Reinhold Remmert: Classical Topics in Complex Function Theory, Graduate Texts in Mathematics, Volume 172, translated from German.
Part A of this outstanding book is dedicated to infinite products. It has six chapters on 140+ pages and covers a lot of classical material, from very basic convergence theory to quite advanced material on functions in one . Highlights include the sine product, partition products, a detailed treatment of the $\Gamma$-function and the $\mathrm{B}$-function, the Weierstraß product theorem, Iss'sa's theorem, Mittag Leffler's theorem and much more. In addition the book has a lot of historical references and remarks and recommendations for further reading.
J. N. Sharma: Infinite Series and Products, see p.129. I did not know about this book, I found it using Google Books - see below.
Online
S. E. Payne: Infinite Sums, Infinite Products, and $\zeta(2k)$
W.W.L. Chen: Fundamentals of analysis Chapter 7
Bibliography for Infinite Products (compiled by John H. Mathews)
Searches
The reason I've included this part is that I only knew about Knopp's book(s) offhand, but it seemed very probable that there are plenty of notes available online. This is how I found Payne's and Chen's notes; you can check the search results for yourself to see, whether you find some other interesting things.
"infinite product" filetype:pdf
"infinite product" absolute filetype:pdf
"infinite product" absolutely filetype:pdf
"infinite product" Cauchy filetype:pdf
"infinite product" absolute at Google Books
"infinite product" Cauchy at Google Books